Limit involving infinite an infinite sum + Probability

In summary, The problem described is a physics problem that involves finding the expectation value of the surface not covered by n pieces of paper when scattered on another piece of paper. The limit of this expectation value as n approaches infinity is also needed. The homework statement provides a hint in the form of a mathematical equation. The attempt at a solution involves defining random variables for the white and black surfaces remaining after scattering n black pieces and calculating the probability of each piece covering a spot or aligning with previous pieces. The next step would be to find the limit of the expectation value, but the calculations presented are incorrect and the individual is seeking guidance.
  • #1
Tomer
202
0
Thanks for reading!

Homework Statement



It's actually a problem in Physics, which is analogical to a problem in probability:
Given are two pieces of paper with area A, one black and one white. we then cut the black one to n pieces, so that each piece has the area A/n (equal sized)
We then let the "little" (n will soon be going to infinity) black pieces fall from above on the other white piece of paper, with the limitation that they have to cover the paper only, and that they either cover "new white spots" on the paper, or align with previously scattered pieces.
I ought to find the expectation value of the surface *not covered* by n pieces.
I then ought to find the limit of this, when n goes to infinity.

Homework Equations



[itex]Lim_{n\rightarrow\infty} (1+\frac{1}{n})^n = e[/itex]
(I've been given that as a hint).

The Attempt at a Solution



I defined S[itex]_{w}[/itex] and S[itex]_{b}[/itex] to be random variables, describing the white and black surfaces remaining after scattering n black pieces.

So:
[itex] P(S_{w} = k\frac{A}{n}) = P(S_{b} = (n-k)\frac{A}{n}) = 1(1-\frac{1}{n})(1-\frac{2}{n})...(1-\frac{n-k-1}{n})(\frac{n-k}{n})^k[/itex]

In the last step I just logically thought of the probability - if I want to cover k-n spots with black pieces using n black pieces, the first piece has probability 1 to cover any spot, the second has probability (1-1/n) to cover any other "free spot", and so on... until the (n-k)'th piece has the probability [itex]1-\frac{n-k-1}{n}[/itex] to cover a new white spot, since n-k-1 pieces are laying there, covering n-k-1 different spots. After "mannaging" to cover n-k black spots, I have to ensure that the remaining k pieces align with previous pieces. So each piece I throw has the probability [itex]\frac{n-k}{n}[/itex] to align with the n-k pieces now on the white paper. k times means [itex](\frac{n-k}{n})^k[/itex]

Assuming this is correct, all I need now is the limit of the expectation value of [itex]S_{w}[/itex], as n goes to infinity.

[itex]Lim_{n\rightarrow\infty} \sum^{n}_{k=1}(k\frac{A}{n})(1-\frac{1}{n})(1-\frac{2}{n})...(1-\frac{n-k-1}{n})(\frac{n-k}{n})^k[/itex]

And this is where I'm stuck. I just don't know how to evaluate this. I'd appreciate tips. Of course, if something in my derivation is wrong I'd appreciate your help in correcting it.

Thanks A-LOT!

Tomer.
 
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  • #2
Ok, I think my calculation is totally wrong.

I'd appreciate any hints as to how to find the probability... :-\
 

1. What is an infinite sum?

An infinite sum is a mathematical concept in which an infinite number of terms are added together to form a sum. It is often used in calculus and other areas of mathematics to represent a value that cannot be obtained by adding a finite number of terms.

2. What is a limit involving an infinite sum?

A limit involving an infinite sum is a mathematical expression that describes the behavior of an infinite sum as the number of terms approaches infinity. It is used to determine the value that an infinite sum "approaches" or gets closer to as more terms are added.

3. How do you calculate a limit involving an infinite sum?

To calculate a limit involving an infinite sum, you would need to determine the general pattern or rule of the sum and then use mathematical techniques such as the limit definition, ratio test, or integral test to evaluate the limit.

4. What is the relationship between limits involving infinite sums and probability?

Limits involving infinite sums can be used in probability to represent the probability of an event occurring when the number of trials or events approaches infinity. This is known as the Law of Large Numbers and is used to demonstrate the convergence of a probability to its expected value.

5. How are limits involving infinite sums used in real-world applications?

Limits involving infinite sums have many real-world applications in fields such as physics, engineering, economics, and computer science. They can be used to model and analyze complex systems and phenomena, such as the behavior of particles in a fluid or the growth of a population over time.

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